Geometer’s Sketchpad®

Running Head: Geometer’s Sketchpad

Geometer’s Sketchpad®: Advancing Middle School Curriculum

Karleen Vogt

Vanderbilt University

CapstoneJune 15, 2011

Geometer’s Sketchpad®

Abstract

This paper will investigate the implementation of a product of Key Curriculum Press®,

Geometer’s Sketchpad® (GSP), into a traditional middle school mathematics curriculum:

Larson’s Middle School Math: Tennessee. GSP, a dynamic and engaging software program,

allows students to be able to explore geometric, algebraic, and statistical concepts through

creation and manipulation of mathematical notions (Hinders, 1992). The students’ ability to

generate their own products and conjectures makes them agents in their own learning (Driscoll,

2007). GSP also leads to effective group work and whole class discussions. By looking at

learners and learning, the learning environment, curriculum and instructional strategies, and

assessment this paper will explore the benefits of implementing GSP in middle school

classrooms. The introduction to this paper will also briefly examine the reform of traditional

curricula and what implications this has for technology. The conclusion of the paper will

provide educators with an understanding of the implementation of GSP in traditional type

curricula and offer suggestions for execution of GSP in the middle school classroom. Four GSP

tasks will be provided and discussed along with images of the completed task in the GSP

software. These four tasks cover geometric, algebraic, and statistical understandings that are

proposed by Tennessee and national standards.

Geometer’s Sketchpad®

Introduction

Curricula across the United States are progressively reforming in response to the focus on

how people learn most effectively and how to improve the structure of acquiring knowledge.

Specifically in mathematics, traditional approaches emphasize direct instruction where the

teacher is the preeminent source of knowledge in the classroom. Additionally, discourse is not

accentuated and concepts are often taught in isolation. Students are shown one standard

technique and expected to reproduce the method or algorithm when necessary. The new

developments in curricula have formed mathematics courses where discovery, discussion, and

connectedness are at the center of instruction. Reform mathematics has focused on including

conceptual understanding and application of mathematics to real life context, ideas often

overlooked in traditional texts. Battista states, “For most students, mathematics is an endless

sequence of memorizing and forgetting facts and procedures that make little sense to them

(1999).” It is for these reasons, among others, that teachers have looked towards restructuring

curricula and instruction to assist in bridging this gap between procedural and conceptual

learning.

Regardless of the reform movement in mathematics, traditional curricula remain the norm

across the United States. Frequently, teachers find themselves working from procedural based

resources. Technology, however, can become a plausible avenue for teachers who still desire to

incorporate engagement, discovery, and independent thinking, the hallmarks of reform

mathematics, into their classroom. The mathematical investigations created through geometry

software give students the opportunity to create and test conjectures, offering an active role in the

learning process. The professor of curriculum and instruction in the College of Education at the

University of Illinois at Urbana, Gloriana González, states that when her students “used dynamic

Geometer’s Sketchpad®

geometry software, they were more successful in discovering new mathematical ideas than when

they used static, paper based diagrams (“Technology & Learning”, 2010).” Furthermore, Keastan

and Sinclair conducted a study that showed 71% of teachers incorporate geometry software to

enhance student understanding of material (2009). Additionally, it was found that these computer

based tasks helped to solve current and persistent difficulties teachers have in delivering the

curriculum. These challenges include addressing misconceptions and providing visual aids,

among other difficulties that teachers sometimes face (Keastan & Sinclair, 2009). Technology

can be utilized as an instructional tool to enhance learning, transfer, and comprehension of

mathematics concepts.

This project will investigate the implementation of a product of Key Curriculum Press®,

Geometer’s Sketchpad® (GSP), into a traditional middle school mathematics curriculum:

Larson’s Middle School Math: Tennessee. GSP, a dynamic and engaging software program,

allows students to be able to explore by creation and manipulation of geometric, algebraic, and

statistical concepts (Hinders, 1992). The students’ ability to create their own products and

conjectures makes them agents of their own learning (Driscoll, 2007). By looking at learners

and learning, the learning environment, curriculum and instructional strategies, and assessment

this project will explore the benefits of implementing GSP in middle school classrooms. The

conclusion of this project will provide educators with an understanding of the implementation of

GSP into traditional type curricula and offer suggestions for execution of GSP in the middle

school classroom. Supplemental material for Larson’s Middle School Math will be created using

GSP, touching upon algebraic, geometric, and statistical concepts. The entire curriculum was

reviewed and key concepts that could be enhanced by GSP were made into tasks that could be

executed during instruction. Step-by-step task directions and images of the GSP task are

Geometer’s Sketchpad®

included. The four tasks that are discussed cover geometric, algebraic, and statistical concepts

that are proposed by Tennessee and national standards.

Before discussing the specific GSP tasks, this paper will address learners and learning,

the learning environment, curriculum and instructional strategies, and assessment through the

general integration of GSP into the classroom.

Learners and learning

No two students reason, develop and learn in the same manner (Milner, 2010). Hence,

there is not one correct way to present material to students. It is vital that in presenting

mathematics to students it is done in multidimensional techniques that simultaneously engage

and apply to students (Milner, 2010). Since technology is a necessary part of our everyday

routine, Geometer’s Sketchpad can act as a catalyst to foster algebraic, geometric, and statistical

habits of mind. This program allows individuals or groups to display, construct, transform, and

measure geometric shapes, algebraic functions, and statistical concepts (Exploring geometry with

The Geometer's Sketchpad, 1992). Once a concept has been explored, one can adapt, drag, and

manipulate images to develop generalizations (Driscoll, 2007). Through guidance, students can

discover aspects of geometry, algebra, and statistics in a hands-on manner. The students’ ability

to create their own products and conjectures gives a sense of independence and accomplishment.

Finally, GSP can address common student errors or naive conceptions (Kasten & Sinclair, 2009).

Through manipulations and tests, students can see mistakes they may make in their reasoning.

It is through appropriately adapting traditional material to incorporate GSP that this paper

will further touch upon the learner and learning. Examples will be given of suitable GSP

activities/tasks that are dynamic, cognitively stimulating, and mathematically relevant.

Geometer’s Sketchpad®

Learning environment

The environment created surrounding the student often determines the motivation the

student has for mathematics (Milner, 2010). A traditional curriculum involves individual work

with minimal teacher-student interaction and limited use of manipulatives. However, in a

computer lab, students are focused on using a tool to inspire learning and can communicate with

both peers and the teacher (Battista, 2002). It should not be ignored that this type of environment

involves creative monitoring. The computers must be set up in such a way that the teacher can

monitor the student’s actions. Putting the computers around the perimeter of the classroom may

allow for proper observation but might be difficult for instruction. Rows, on the other hand, may

be ideal for instruction but pose an issue for keeping students on task. Therefore, other lab

management practices need to be applied appropriately. For example, when the teacher is

speaking, he/she could ask the students to put their hands on the top of their heads to discourage

working during instruction. Additionally, teachers should make sure certain websites are

unavailable for students during lab hours.

Building upon the interactive classroom, GSP tasks are conducive to engaging in high

level mathematical discussions (Battista, 2002). Since students are manipulating and

transforming geometric, algebraic, and statistical figures, they can use their work to form

conjectures and justify their findings (Driscoll, 2007). Not every property is proven or supported

in the same fashion. Discussions of similarities and differences among approaches can also be

highlighted (Battista, 2002).

It is through appropriately adapting traditional material to incorporate GSP that this paper

will further touch upon the learning environment. Examples will be given of suitable GSP

activities/tasks that create an engaging atmosphere centered on discourse and discussion.

Geometer’s Sketchpad®

Curriculum and instructional strategies

Given any middle school program, GSP can be incorporated as supplemental material.

Additionally, GSP fits the widespread curriculum that is often found in 7 th and 8th grade because

it reaches to geometric, algebraic, and statistical concepts (KCP Technologies, 2009). GSP may

work best in 60-90 minute class period so that the students get enough time to discover or

practice the topic of study. Teachers can create interactive and dynamic activities which aid

students in their own exploration of concepts, thus making the students’ learning more notable

(KCP Technologies, 2009). Additionally, it can be utilized as a demonstration tool or to review a

process learned in geometry, algebra, or statistics. GSP tasks support small group or individual

learning, along with reaching students at all levels of thinking.

GSP can complement effective curriculum designs such as those offered in

Understanding by Design (Wiggins & McTighe, 2005). In the framework presented, essential

questions are the central focus of a unit. According to Wiggins and McTighe an essential

question is “deemed essential” if it is not easily answered, can “get to the heart” of a topic, and

can be found outside of the classroom in everyday life (2005, p.107). GSP creates many open-

ended opportunities through which everyday life experiences can be explored. Additionally, the

software can be utilized for algebraic investigations, geometric problem solving, statistical

inquiries, among other mathematical concepts.

It is through appropriately adapting traditional material to incorporate GSP that this paper

will further touch upon the curriculum and instructional strategies. Examples will be given of

suitable GSP activities/tasks that are interactive and support top curriculum designs.

Geometer’s Sketchpad®

Assessment

The hands-on interactive experience fostered by GSP provides a solid understanding of

mathematical concepts and, thus, demonstrating a student’s ability to better recall information

during assessments. Not only does GSP prepare for summative assessments, but it can be used as

a pre-assessment and formative assessment. Teachers may create GSP tasks that act in way to

discover what the students understand before beginning a topic. Furthermore, as students work

through tasks, they can self-reflect on the particular concepts being studied. While working on a

task or after completion, students may insert a text box and include a summary about how they

completed their construction and what they learned from this construction. Also during

construction, teachers can gauge student comprehension of manipulations and justifications by

asking appropriate pressing questions (Driscoll, 2007). The software allows for more indepth

questioning because the tasks are complex, often with multiple solutions. GSP tasks can be

printed out or saved on the computer for future reference. Finally, teachers may also choose to

give a summative assessment in an engaging way using GSP.

It is through appropriately adapting traditional material to incorporate GSP that this paper

will further touch upon assessment. Examples will be given of suitable GSP activities/tasks that

can act as pre-assessments, formative assessments, and/or summative assessments.

Tasks

An overview of each task will be presented along with an explanation of its role in

creating a more productive learning experience. Instructions for implementing the task and

images of the GSP file will be included.

Geometer’s Sketchpad®

Task 1

This task could be completed in conjunction with Section 11.7: Slope-intercept form,

page 577, in Larson’s Middle School Math: Tennessee. In order to be successful with this

activity, students should understand slope and recognize lines with positive, negative, undefined,

and zero slopes (Tucker, 2010). Minimal experience with GSP would not reduce learning. The

task is completed as follows:

Investigating Slopes lab (Adapted from Tucker, 2010)

1. Start The Geometer’s Sketchpad.

2. Draw a horizontal line segment across the center of the screen.

3. Find the midpoint of the line. Construct a circle from the midpoint that reaches the endpoints ofthe line segment.

4. Label the intersection of the line segment and the circle as points A and B.

5. Place six points at various places on one semicircular arc of the circle.Label the points C through H.

6. Connect each of these six points to the points A and B.

7. Measure the slopes, to the nearest hundredth, of each pair of line segments meeting at the sixpoints on the semicircle. Record the slopes in the table below.

8. Measure the angles formed at each of the six points on the semicircle. Describe the anglemeasurements here: __________________________________________.

9. Use the GSP calculator to calculate the entries for the columns in the table.

PointName

m1, Slope to A m2, Slope to B m1 + m2 m1 - m2 m1 . m2 m1 ÷ m2 1/m1 1/m2

CDEFGH

A B

Midpoint

Geometer’s Sketchpad®

10. Look at the entries in the columns in the table. See if there are patterns among any of the valuesof the sum, difference, product, quotient, or reciprocals of the slopes of the line segments. Justifyyour answer.

11. What is the vocabulary word used to describe two lines that meet at the angle you measured instep 8?

12. Formulate a conjecture about the slopes of these lines. Justify your answer.

This particular task can prove to be a very powerful tool when investigating perpendicular

lines. Using the built in calculator can ensure accuracy and save time performing tedious

calculations. Students are able to quickly and precisely create perpendicular lines and 90° angles

(Tucker, 2010). Tucker states, “The inscribed right triangle forces perpendicularity wherever the

student places the point on the semicircle. In contrast with a paper exercise, here the accuracy of

the angle measure is automatic (p. 606).” As a result, students are taking less time to complete

the construction and more time analyzing patterns from the results (Tucker, 2010). Instead of

memorizing facts about perpendicular lines, students are able to detect, observe and discover

rules, forming a more solid understanding of why these theories exist.

In this task, students are introduced to more than concepts surrounding perpendicular

lines. It can be proven that any angle inscribed in a semi-circle is a right angle. As students are

determining correlations between perpendicular lines, they are utilizing this fact. This can be

brought up during this discussion or can be revisited when angles and chords are investigated to

connect mathematical concepts.

The main goal of this lesson is for the students to learn that the sum and difference vary

but to see that the product is always -1 (Tucker, 2010). Additionally, students can determine that

Slope AC( )Slope BC( )

= -13.58

Slope AC( )⋅ Slope BC( ) = -1.00

Slope AC( )- Slope BC( ) = 3.96

Slope AC( )+ Slope BC( ) = 3.41

m∠AHB = 90.00°m∠AGB = 90.00°m∠AFB = 90.00°m∠AEB = 90.00°m∠ADB = 90.00°m∠ACB = 90.00°

Slope BH = -5.29Slope BG = -2.89Slope BF = -1.25Slope BE = -0.72Slope BD = -0.61Slope BC = -0.27

Slope AH = 0.19Slope AG = 0.35Slope AF = 0.80Slope AE = 1.39Slope AD = 1.65Slope AC = 3.69

AB

C

DE

F

G

H

Geometer’s Sketchpad®

perpendicular lines have reciprocal slopes. Along with building conceptual knowledge, this task

introduces many features of GSP and allows students to obtain more skills in using the software.

This particular task creates a powerful learning environment through discovery and

discussion. In small groups or in a whole class discussion, the students can draw conclusions

from data that they have collected. The students are prompted to investigate more measurements

than needed to deduce the pattern (Driscoll, 1999). They must sort through the information,

determining what is significant and what does not form a pattern. A student might argue for a

pattern when adding m1 and m2, while another student might not be convinced of it. This allows

the students to form informed opinions and support their findings.

Task 2

This task could also be completed in conjunction with Section 11.7: Slope-intercept form,

page 577, in Larson’s Middle School Math: Tennessee. In order to be successful with this

activity, students should know what a function is, including input and output relationships.

Additionally, they should be somewhat familiar with GSP.

Transforming Linear Equations in Slope-Intercept Form

1. Open Geometer’s Sketchpad. Under Graph on the main tool bar, click define coordinatesystem (Make sure the grid form is in square grid or rectangular grid. Also, you can hidegrid and just view the axes.).

2. Under Graph, click plot new function (Ctrl + G).

3. Under Equation, make sure y-notation is checked. To type in the function y = x, simplyput an x in the white box. Notice the equation is shown above (some form of f(x) = x).Click okay.

Geometer’s Sketchpad®

4. Positive coefficient: Also graph y = 2x and y = x on the coordinate system. What canyou conclude about positive coefficients in respect to transforming linear equations inslope-intercept form? Justify your response. Note: At any time you want to delete a function, highlight that function only and

click the delete button.

5. Negative coefficient: Graph y = -3x and y = x on the coordinate system. What canyou conclude about negative coefficients in respect to transforming linear equations inslope-intercept form? Justify your response.

6. Positive Constant: Graph y = x + 3 and y = x + 1.5 on the coordinate system. What canyou conclude about positive constants in respect to transforming linear equations inslope-intercept form? Justify your response.

7. Negative Constant: Graph y = x - 4 and y = x - on the coordinate system. What canyou conclude about negative constants in respect to transforming linear equations inslope-intercept form? Justify your response.

Geometer’s Sketchpad®

8. Practice: Anticipate the graph of the given function. Include a reasoning or justificationfor your graph. Use Geometer’s Sketchpad to graph the equation and check your work.

Equation Anticipated Graph Justification/Reasoning Actual Graph

y = - 4x + 7

y = x + 1

y = 5x

y = - x – 2

y = 4

What happens in the final graph? Why? Justify your answer.

14

13

12

11

10

9

8

7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

-11

-12

-13

-8 -6 -4 -2 2 4 6 8 10 12 14

w x( ) = x

v x( ) = x-13

u x( ) = x-4t x( ) = x+1.5s x( ) = x+3

r x( ) = -5⋅x2

q x( ) = -2⋅x

h x( ) = 2⋅x3

g x( ) = 2⋅x

Geometer’s Sketchpad®

This task allows students to investigate algebraic expressions along with their graphical

representation. In groups or individually, students can explore how negative and positive

coefficients result in positive or negative slopes. The students will see how this changes the basic

y=x graph. In addition, students will make generalization about the role of the y-intercept in the

algebraic expression and graphical representation. Finally, after students identify patterns, they

must anticipate other examples of algebraic expressions. They are even required to make a

conjecture about the line y = 4.

In this task, the software can provide a way for the students to investigate algebraic

expressions without focusing on the intricate particulars of graphing equations. The students are

able to quickly graph the expression and gain immediate feedback through the visual. They do

not have to spend excessive amounts of time on the graphing portion of the task and are instead

able to concentrate on identifying patterns, developing explanations, creating conjectures, and

forming generalizations (Driscoll, 199).

A strength of this GSP task is the visual support for transformations (Driscoll, 2007). The

students read the algebraic expression and are asked to graph it using the function tool. They are

then able to compare and contrast to a known function, creating a hypothesis of the patterns that

they see. Additionally, students are required to support their reasoning for their anticipated

graphs. This will hopefully allow the students to gauge their own comprehension and write about

mathematical relationships. The students are making connections among multiple

representations, reinforcement their understanding of the material.

Task 3

J Tour 2: A Theorem About Quadrilateralssometimes students will have trouble remembering or understanding aparticular theorem. Discovering the theorem themselves or actively exploringits consequences can make a huge difference in their level of recal-andunderstanding. Thjg tpur challenges you to discover a theorern. As you solvethe challenge, you'll learn many new Sketchpad features.

)9;v tnal you have a quadrilateral, recall the theorern you're exploring, it begins"when the midpoinfs of the sides of a quadrilateral are connected. . . .,, you willcontinue by constructing the four midpoints and then connecting them.

6. Deselect all objects by clicking in blank space. select the four sides of thequadrilateral in clockwise or iounter-cloikwise order.

\r'"After step 3

A/ \<r)

After step

When the miilpoints of the sides of a quadilateral are connected, the resultingshape is alrnays a _.

What You Will Learno How to construct a polygon using the Segment tool

o How to show an object's label

o F{ow to measure lengths and angles

. How to create a caption

o How to apply formatting options to text

The theorem stated above has a key word left blank. Our goal is to use Sketchpadto discover what word fills the blank. Don't worry if this theorem is unfamiliar91, ol the other hand, if it's too simple. The point is to leam about how to useSketchpad for making and testing conjectures.

Constructing a General Quadrilateral

1. start sketchpad if it isn't already running. If it is running, choose NewSketch from the File menu.

2. Use the Segment tool to draw a segment.3. Construct a second segment that shares one endpoint with

the first.

4. Construct two more segments to complete thequadrilateral.

5. using the Arrow toof drag some of the points and segments tomake sure the figure holds together-does it pass the?rag test?

Constructing the Midpoint euadritateral

A "general I

quadrilateral" is a Iquadrilateral with no I

special properties, I

such as congruent Isides or right angles. I

@ 2002 Key Curriculum press The Geometer's Sketchpad Workshop Guide . S

7. Choose Midpoints from the Construct menu.

8. Wiih the four midpoints still selected, choose Show Labels from theDisplay menu.

9. With the four midpoints sflll selected, choose Segments fromthe Construct menu.

10. Use the Arrow tool to drag each point in your figure. Especiallymake sur€ that the midpoints are really midpoints. (Can you

" drag them so thatthey arenot?)

Dragging and Measuring to Confirm Your Conjecture

You mav alreadv have a hvpothesis about what kind of ouadrilaterai the innerfigure is. Be careful, thoughl because it may be that shape only for a parficularshape of the outer quadrilateral. You need to convince yourself that yourconjecture holds no matter what shape the outer quadrilateralis. This requiresdragging. To gain even more confidence in your answer, you'll takemeasurements as well.

11. Drag various parts of the original quadrilateral around, keeping an eye onthe inscribed figure.

12. Now, select the four inner segments and choose Lengthfrom the Measure menu.

1l-"1::':"li-9 !!':: F 13. Deselect all objects. Then select three consecutivepoints to measure an l -"'

-:-;",:-'::::'-:?:::' :l':l::;-.. 'l-:::';;:::::-l':-

,--. m/.ABC =84.61';;; ffi;;il: I

midpoints and choose Angle from the Measure menu'the second selected I t a. 5g1sgt three consecutive midpoints again, with a differentpoint is the vertex' I midpoint as the second selected point (vertex), and choose Angle. Repeat

this step two more times, making sure a different midpoint is the secondpoint selected each time.

15. Choose Calculate from the Measure menu.Click on the measure for /.ABC, then clickon the + sign (or type it on the keyboard),then click on the measure for IBCD, then click OK. What does this tell us?

16. Drag various parts of the original quadrilateral, making it wide, skinny,concave, and so on. \Mhat happens to the measurements? \zVhat changes, andwhat stays the same? Are you more confident of your conjecture now? Asconfident as you may feel, do you think you've prouen the theorem?

\

You can also use the [>Text tool to show the I

labels. Cl ick on an Iobject with the Text I

tool to display (or Ilater hide) its label. I

Double-click the Il^h^l : r^^la +^ |r o u € r r ( D E ' r ( u

Ichange the label. I

ATA'#=€After step 9

m AE = 1.15 cm

nIABC +nZBCD = 180.00"

* Diagonals bisecteach other.* Opposite interiorangles arecongruent thusparallel sides*Interior anglesequal 360 degrees

When the midpoints of the sides of aquadrilateral are connected, the resultingshape is always a parallelogram.

m∠ABC+m∠BCD = 180.00°m∠CDA = 116.38°m∠BCD = 63.62°

m∠BAD = 63.62°m∠ABC = 116.38°

m DA = 4.94 cmm CD = 8.69 cmm BC = 4.94 cmm AB = 8.69 cm

O

D

C

B

A

Geometer’s Sketchpad®

This task could be completed in conjunction with Section 8.3: Quadrilaterals, page 386,

in Larson’s Middle School Math: Tennessee. In order to be successful with this activity, students

should have some familiarity with properties of quadrilaterals and parallelograms, as well as with

tools available in GSP. During the task, students will acquire additional techniques for

effectively utilizing GSP.

Here the students discover that when the midpoint of the sides of a quadrilateral is

connected, the resulting shape is always a parallelogram. In most classrooms, this would be

among a list of properties concerning quadrilaterals that would have to be “committed to

memory”. Memorizing a theorem does not usually result in comprehension, nor allow for future

recall. Discovering the theorem or actively exploring it can increase recall and understanding.

When students are involved in “learning lists of properties is not nearly as important as being

involved in the processes of developing and using a property-based conceptual system for

reasoning about these shapes (Battista, 2002, p. 333).” In addition, it allows the students to take

an abstract concept and make it more concrete (Battista, 2002). Similar to the task Investigating

Slopes, students discover the answer to a theorem. In this task they explore the question, “When

the midpoint of the sides of a quadrilateral is connected, the resulting shape is always a

_______________? (Adapted from The Geometer’s Sketchpad® Workshop Guide)”

Geometer’s Sketchpad allows for differentiation among learners and acts as a support

for higher order thinking. This particular task lends well to many differentiation strategies. The

first is that the task supports learners at any level of geometric reasoning. Van de Walle proposed

a five-level hierarchy of understanding spatial thought comprising level (0) Visualization, (1)

Analysis, (2) Informal Deduction, (3) Deduction, and (4) Rigor (Van de Walle, 2007). Most

middle school students would be at a level 1, 2, or 3. During this task, students form a

Geometer’s Sketchpad®

quadrilateral and are asked to investigate if it passes the drag test. With this investigation

technique, students can begin to move from the level in which they are thinking to a higher level

(Hinders, 1992). They can begin to explore the “all” case through the drag test or develop

relationships around investigating properties. In the end, the students have to also prove and

justify the results of their conjecture concerning the theorem. This allows each student to enter

into the problem at their own level and utilize the parts of the software they see fit. In addition,

there are ‘Further Challenges’ at the end of the section that could be utilized for higher level

achieving students. These challenge problems both help students to organize their thoughts more

effectively to create a stronger proof and allow students to extend their thinking to other shapes

(The ‘Further Challenge’ are not included in this document but may be added for advanced

students).

Specifically, this task also creates dynamic learning for the students. The students

display, construct, transform, and measure geometric shapes (Exploring geometry with The

Geometer's Sketchpad, 1992). As touched upon previously, once the quadrilateral or

parallelogram is created, one can adapt, drag, and manipulate the image to develop

generalizations (Driscoll, 2007). Through guidance, the students discover a theorem pertaining to

all quadrilaterals in a hands-on manner. Finally, the students’ ability to create their own products

and conjectures gives a sense of independence and accomplishment. They are not simply

listening to the teacher pour information into their brains. The learning environment is

transformed into a place where the students hold the knowledge and can access it with through

technology.

Task 4

Exploring Algebra 2 with The Geometer’s Sketchpad 8: Probability and Data 217© 2007 Key Curriculum Press

Box and Whiskers

The box-and-whiskers plot (sometimes just called a box plot) is a recent development

in statistical analysis. You cannot derive any detailed information from it, but it

gives you a convenient, easily understood graphical representation of the data

distribution.

SKETCH AND INVESTIGATE

1. Open Box and Whiskers.gsp.

The sketch contains ten data values represented as points on parallel lines.

Above the points are a box and whiskers. You can change a value by sliding its

corresponding point right or left. The data are ordered and displayed on the left,

but the actual numerical values are not important for this activity.

2. Before answering any of the questions, take a minute to experiment with the

sketch. Drag the data points and observe the effect.

Each of the following questions suggests a special shape for the box and whiskers.

In each case, state whether it is possible. If it is not possible, explain why not. If it is

possible, make a rough sketch of the data points that will create that confi guration,

and suggest a real data set that might make this happen. The fi rst one is done as an

example.

Q1 Can one whisker have zero length?

A1 This will occur if the lower one-fourth of the data points all have the same

value. This might happen if there is a lower limit to the data range. Test a group

of people to see how far they can throw a heavy weight. Those who cannot even

lift the weight will all score zero.

Q2 Can the median fall outside of the box?

Q3 Can the box have zero width?

218 8: Probability and Data Exploring Algebra 2 with The Geometer’s Sketchpad© 2007 Key Curriculum Press

Box and Whiskerscontinued

Q4 Can both whiskers have zero length?

3. Press the Show Mean button. The mean is represented by a green bar. Normally,

the mean is not shown on a box-and-whisker plot. It appears here so that you

can observe its relationship to the data distribution.

Q5 Can the mean fall outside of the box?

x

Q6 Can the mean be greater than the maximum?

x

Questions Q7–Q9 involve moving data points. In doing so, you are actually moving

from one data set to another. In real life this could happen when a restaurant

manager changes the prices of certain menu items, or when a sports team makes a

player trade.

Q7 By moving one or more data points, can you move the mean without changing

the box, the whiskers, or the median?

x

x

Q8 Can you move a single data point without changing the mean?

Q9 Can you move two data points without changing the mean?

Q10 Can you change the median by moving a single data point?

Exploring Algebra 2 with Sketchpad(C) 2007 by Key Curriculum Press

Box and Whiskers

Q3 = 34.77

Q1 = 23.52

mean = 29.47

median = 30.42

x10 = 38.46

x9 = 37.11

x8 = 34.77

x7 = 31.44

x6 = 31.19

x5 = 29.64

x4 = 28.17

x3 = 23.52

x2 = 21.43

x1 = 18.93

Show Mean

Geometer’s Sketchpad®

This task could be completed in conjunction with Section 12.2: Box-and-Whisker Plot,

page 601, in Larson’s Middle School Math: Tennessee. In order to be successful with this

activity, students should have been taught quartiles and extreme values, in addition to the

procedure for making a box-and-whisker plot. They should have done some work on interpreting

the plot and using it to display data. This task was taken from Exploring Algebra 2 with The

Geometer’s Sketchpad written in 2007.

One way to foster statistical thinking is through balancing exploration and reflection of

concepts. Mark Driscoll emphasizes this approach to nurture geometric thinking, arguing for the

importance of trying various ways to approach a problem and regularly reflecting upon what is

being learned (Driscoll, 2007). This approach for learning geometry may also be applied to this

statistical task. Balancing exploration and reflection aids students in enhancing their problem

solving techniques, as well as in evaluating the productivity of their investigation and the

direction of the next moves. Using GSP to explore box-and-whisker plots emphasizes this trial

and error approach. The students are able to quickly move around the data to explore things such

as where the mean can fall in the data and on the box-and-whisker plot. They are able to compare

the visual representation with the data.

This task can be used as either an exploratory tool or an assessment tool. As an

exploratory tool, the students would follow the instructions as they have done with the other

tasks. Through this type of implementation, teachers can gauge student understanding through

their manipulations and conjectures (Driscoll, 2007). Additionally, the teacher can ask probing

questions and engage in high level discussions with the students to assess their knowledge and

gain an understanding of their nature of problem solving (Holyoak, 1995).

Geometer’s Sketchpad®

However, the students could also complete this individually or with a partner, as an end

of the unit assessment. The unit may have talked about the sizes of the whiskers, where the mean

can fall, and what affects the medium (for example). However, it might not have been the

explicitly taught. If the students can reason through these small tasks, it would show that they

understand the parts of a box-and-whisker plot. Since they can manipulate the data and view the

resulting changes on the box-and-whisker plot, the students need to be able to justify their

reasoning to each part. That is, they may draw the image or write down values of the data points.

This immediate feedback, given through the GSP software, allows this task to be an effective

pre- or post-assessment.

Conclusion

Given the effectiveness of GSP in fostering reasoning and sense-making in mathematics,

the tasks can be time consuming to implement, and students may not be accustomed to

investigating mathematics in this way. Additionally, the software can be expensive to purchase,

difficult to install, and takes time to learn. Teachers need to be prepared to manage group work

and discussion in their classroom. Despite these few limitations, GSP enhances the learning and

understanding of mathematics.

Geometer’s Sketchpad is effective first because of its influence on the learner. Students

are motivated through the use of technology and being agents of their own learning. GSP reaches

students at every level of mathematical thought and provides the necessary scaffold towards a

more sophisticated understanding. Additionally, GSP demands higher order thinking and

justification of conjectures. Finally, students are able to remember and recall what they learn

Geometer’s Sketchpad®

because GSP helps students to actively organize their thoughts, connect to previous learned

knowledge, and balance exploration with reflection (Driscoll, 2007).

Also, GSP helps to create a more dynamic learning environment through highlighting big

ideas and essential questions (Driscoll, 2007; Wiggins & McTighe, 2005). Its open-ended

methods and justifications support overarching themes. Furthermore, instead of a traditional

class where students sit in rows and the teacher is the source of knowledge, students are

collaborating with each other and the teacher to draw their own conclusions and support their

conjectures. Mathematical discourse is a direct result of GSP tasks, which strengthen

comprehension of the concept at hand.

In addition to using GSP as a pre- or post-assessment tool, assessment of student learning

occurs in day-to-day implementation of GSP. When students utilize GSP, teachers can gauge

their understanding of their actions, manipulations, and justifications while problem solving

(Driscoll, 2007). Through questioning, teachers can gain insight into student progress of

mathematical connections.

As this country tends toward national standards and expectations, GSP is a powerful tool

that helps all curricula align with the principals outlined in the NCTM standards. Geometer’s

Sketchpad:

(1) Emphasizes mathematical understanding, problem solving, and reasoning;

(2) Provides all students with opportunities to learn important mathematical ideas andskills;

(3) Provides opportunities for significant interactions of students and teachers, using avariety of communication methods (e.g., writing, explaining, presenting, defending);

(4) Incorporates the use of technology that enhances understanding; and

Geometer’s Sketchpad®

(5) Uses various forms of assessment to better gauge student understanding and toimprove instructional practice (Rey et al, 1999).

These are deemed to be important goals by the NCTM and many educators. Geometer’s

sketchpad helps to obtain these goals. For the reasons presented in this paper, it should be a

resource used by educators across the country.

In all future mathematics education preparation, teachers should be made aware of the

conceptual framework of mathematical tasks and how GSP aligns with its goals and objectives.

The mathematical tasks framework, thought to be an effective way to teach mathematics, has a

Task as Set-up, Implementation, and Discussion phase (Stein, Grover, & Henningsen, 1996). The

task as set-up phase is a time where the teacher and students discuss the context and

mathematical relationships, without lowering the cognitive demand of the task. During the

implementation stage, students are discovering or investigating the concept. Often this is

accomplished through teacher questioning and group work. Finally, in a whole-class discussion,

students show strategies and include justifications for work. Collectively, clear connections

among ideas are established. Strong tasks “are characterized by features such as having more

than one solution strategy, as being able to be represented in multiple ways, and as demanding

that students communicate and justify their procedures and understandings in written and/or oral

form” (Stein, Grover, & Henningsen, 1996, p. 456). The GSP tasks found in the paper support

the above framework and the description of strong mathematical tasks. They provide an

opportunity for discovery and lend well to whole class discussions. Knowing this framework

would help educators acknowledge the importance of using GSP in their classroom.

Geometer’s Sketchpad®

Finally, future curriculum creators can use Geometer’s Sketchpad to enhance their

textbooks so that deeper mathematical relationships are made, students are able to recall more

mathematics, and there is a connection between content and technology. GSP should not

completely replace current mathematics curricula, but be used as a supplemental resource to

develop strong mathematical discourse.

Geometer’s Sketchpad®

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